The graded-graph path-count conjecture

Let $G$ be a graded graph of height $n$, and let $K(G)$ be the number of oriented paths that pass through every level of $G$. Let $X(\sigma)$ be independent, identically distributed random variables with common law $\mu$, and let $B\subseteq\mathbb{R}^n$. Write $P(B;G)$ for the probability that the labels along a path in $G$ produce a vector in $B$.

Graded-graph path-count conjecture. One has

$$1-P(B;G)\geq\left(1-\mu^n(B)\right)^{K(G)}.$$

This conjecture extends the corresponding inequality for trees to arbitrary graded graphs. It asserts that the number of complete oriented paths alone gives the same lower bound for the probability that no path produces a vector in $B$; its resolution is not specified in the source.